MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hbralrimi Structured version   Visualization version   GIF version

Theorem hbralrimi 2936
Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). This theorem contains the common proof steps for ralrimi 2939 and ralrimiv 2947. Its main advantage over these two is its minimal references to axioms. The proof is extracted from NM's previous work. (Contributed by Wolf Lammen, 4-Dec-2019.)
Hypotheses
Ref Expression
hbralrimi.1 (𝜑 → ∀𝑥𝜑)
hbralrimi.2 (𝜑 → (𝑥𝐴𝜓))
Assertion
Ref Expression
hbralrimi (𝜑 → ∀𝑥𝐴 𝜓)

Proof of Theorem hbralrimi
StepHypRef Expression
1 hbralrimi.1 . . 3 (𝜑 → ∀𝑥𝜑)
2 hbralrimi.2 . . 3 (𝜑 → (𝑥𝐴𝜓))
31, 2alrimih 1740 . 2 (𝜑 → ∀𝑥(𝑥𝐴𝜓))
4 df-ral 2900 . 2 (∀𝑥𝐴 𝜓 ↔ ∀𝑥(𝑥𝐴𝜓))
53, 4sylibr 222 1 (𝜑 → ∀𝑥𝐴 𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1472  wcel 1976  wral 2895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727
This theorem depends on definitions:  df-bi 195  df-ral 2900
This theorem is referenced by:  ralrimi  2939  ralrimiv  2947  bnj1145  30149
  Copyright terms: Public domain W3C validator